Also, by the last article, {4 3}{5 7} = {3 4}{7 5} ={3}{7}{4}{5} {4}{5} {3}{7}= {3}{7} {4}{5} The reasoning is clearly general, and we may now say that a X b= b X a, where a and b are any positive quantities, integral or fractional.
The same is true for any number of quantities, hence the factors of a product may be taken in any order. This is the Commutative Law for Multiplication.
38. Again, the factors of a product may be grouped in any way we please. Thus abed = axbxcxd = (ab) X (cd) = a X (be) x d = a x (bed). This is the Associative Law for Multiplication.
39. Since, by definition, a^ = aaa, and a^ = aaaaa,
a^ \times a^ = aaa x aaaaa = aaaaaaaa = a^ = a^+^ ; that is, the index of a letter in the product is the sum of its indices in the factors of the product. This is the Index Law for Multiplication.
Again, 5 a^2 = 5 aa, and 7 a^3 = 7 aaa. 5 a^2 X 7 a^3 = 5 X 7 X aaaaa = 35 a^5.
When the expressions to be multiplied together contain powers of different letters, a similar method is used.
Ex. 5 a3b2 X 8 a2x^3 = 5 aaabb x 8 aabxxx = 40 a^5b^2x^3.
Note. The beginner must be careful to observe that in this process of multiplication the indices of one letter cannot combine in any way until those of another. Thus the expression 40a^b^x^ admits of no further simplification.
